Hi Jamie,
Cycling will cause a muscle called opponens pollicis to go into a repetitive strain injury because you are holding onto the handle bars for long periods of time. You can see the muscle if you go to the section titled "Anatomy Lessons" on my website
http://www.carpaltunneltreatment.org/.
Aside from the muscles you mentioned, the opponens pollicis draws your thumb in toward your palm, and when you are cycling it is not only held in this position for hours, but it is also using force to maintain control of the handles. Since the muscle originates on the bridge to the carpal tunnel (flexor retinaculum) and inserts into the base of your thumb, when it goes into a contraction or spasm you will feel it in your wrist and thumb.
Working on the opponens pollicis will release the tension, and it's easy to do. Do you have my book
Treat Yourself to Pain-Free Living? Or do you have the Julstro System for hand/wrist pain? They both teach you how to treat the thumb muscles.
I think it's also important for you to work on all of the flexor muscles because they are gripping your hand into a fist while you cycle, so they are shortening the whole time. Your biceps and the pectoralis minor are also being repetitively strained while cycling, so it would be good if you work on them too.
Fortunately it's easy to do all of the treatments, and they work really well. I don't think you have anything to be concerned about, it's just a matter of some focused time and energy and you'll be fine.
Wishing you well,
Julie